Hands-on Exercise 8: Hedonic Pricing model with Geographically Weighted Regression

Show the code
pacman::p_load(sf, tidyverse,tmap,olsrr,corrplot,ggpubr,spdep,GWmodel,gtsummary)

Import Data

This exercise focuses on building a Hedonic pricing model on condo prices in 2015 with Geographically weighted model.

Geospatial Data Wrangling

Through observation, the imported shapefile is not in the expected coordinate system - SVY21, need to apply st_transform() on the data to convert to the correct coordinate system.

Show the code
mpsz <- st_read(dsn = "../../data/geospatial",
                layer = "MP14_SUBZONE_WEB_PL") %>% 
  st_transform(3414)
Reading layer `MP14_SUBZONE_WEB_PL' from data source 
  `/Users/tangtang/Desktop/IS415 Geospatial Analytics and Applications/practice/is415gaa/data/geospatial' 
  using driver `ESRI Shapefile'
Simple feature collection with 323 features and 15 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: 2667.538 ymin: 15748.72 xmax: 56396.44 ymax: 50256.33
Projected CRS: SVY21
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st_crs(mpsz)
Coordinate Reference System:
  User input: EPSG:3414 
  wkt:
PROJCRS["SVY21 / Singapore TM",
    BASEGEOGCRS["SVY21",
        DATUM["SVY21",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["degree",0.0174532925199433]],
        ID["EPSG",4757]],
    CONVERSION["Singapore Transverse Mercator",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["northing (N)",north,
            ORDER[1],
            LENGTHUNIT["metre",1]],
        AXIS["easting (E)",east,
            ORDER[2],
            LENGTHUNIT["metre",1]],
    USAGE[
        SCOPE["Cadastre, engineering survey, topographic mapping."],
        AREA["Singapore - onshore and offshore."],
        BBOX[1.13,103.59,1.47,104.07]],
    ID["EPSG",3414]]

We can see the size of the boundary in terms of pixels using st_bbox(). It will return the size of the plot in Cartesean coordinates (which truly reflects that of Singapore).

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st_bbox(mpsz)
     xmin      ymin      xmax      ymax 
 2667.538 15748.721 56396.440 50256.334 

After we are done with pre-processing geospatial data, we need to import aspatial data used in this exercise, which is condo prices in 2015.

Show the code
condo_resale <- read_csv("../../data/aspatial/Condo_resale_2015.csv")
Rows: 1436 Columns: 23
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
dbl (23): LATITUDE, LONGITUDE, POSTCODE, SELLING_PRICE, AREA_SQM, AGE, PROX_...

ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
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glimpse(condo_resale)
Rows: 1,436
Columns: 23
$ LATITUDE             <dbl> 1.287145, 1.328698, 1.313727, 1.308563, 1.321437,…
$ LONGITUDE            <dbl> 103.7802, 103.8123, 103.7971, 103.8247, 103.9505,…
$ POSTCODE             <dbl> 118635, 288420, 267833, 258380, 467169, 466472, 3…
$ SELLING_PRICE        <dbl> 3000000, 3880000, 3325000, 4250000, 1400000, 1320…
$ AREA_SQM             <dbl> 309, 290, 248, 127, 145, 139, 218, 141, 165, 168,…
$ AGE                  <dbl> 30, 32, 33, 7, 28, 22, 24, 24, 27, 31, 17, 22, 6,…
$ PROX_CBD             <dbl> 7.941259, 6.609797, 6.898000, 4.038861, 11.783402…
$ PROX_CHILDCARE       <dbl> 0.16597932, 0.28027246, 0.42922669, 0.39473543, 0…
$ PROX_ELDERLYCARE     <dbl> 2.5198118, 1.9333338, 0.5021395, 1.9910316, 1.121…
$ PROX_URA_GROWTH_AREA <dbl> 6.618741, 7.505109, 6.463887, 4.906512, 6.410632,…
$ PROX_HAWKER_MARKET   <dbl> 1.76542207, 0.54507614, 0.37789301, 1.68259969, 0…
$ PROX_KINDERGARTEN    <dbl> 0.05835552, 0.61592412, 0.14120309, 0.38200076, 0…
$ PROX_MRT             <dbl> 0.5607188, 0.6584461, 0.3053433, 0.6910183, 0.528…
$ PROX_PARK            <dbl> 1.1710446, 0.1992269, 0.2779886, 0.9832843, 0.116…
$ PROX_PRIMARY_SCH     <dbl> 1.6340256, 0.9747834, 1.4715016, 1.4546324, 0.709…
$ PROX_TOP_PRIMARY_SCH <dbl> 3.3273195, 0.9747834, 1.4715016, 2.3006394, 0.709…
$ PROX_SHOPPING_MALL   <dbl> 2.2102717, 2.9374279, 1.2256850, 0.3525671, 1.307…
$ PROX_SUPERMARKET     <dbl> 0.9103958, 0.5900617, 0.4135583, 0.4162219, 0.581…
$ PROX_BUS_STOP        <dbl> 0.10336166, 0.28673408, 0.28504777, 0.29872340, 0…
$ NO_Of_UNITS          <dbl> 18, 20, 27, 30, 30, 31, 32, 32, 32, 32, 34, 34, 3…
$ FAMILY_FRIENDLY      <dbl> 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0…
$ FREEHOLD             <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1…
$ LEASEHOLD_99YR       <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
Show the code
head(condo_resale$LONGITUDE) #see the data in XCOORD column
[1] 103.7802 103.8123 103.7971 103.8247 103.9505 103.9386
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head(condo_resale$LATITUDE) #see the data in YCOORD column
[1] 1.287145 1.328698 1.313727 1.308563 1.321437 1.314198

Summary statistics of condo resales prices in 2015

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summary(condo_resale)
    LATITUDE       LONGITUDE        POSTCODE      SELLING_PRICE     
 Min.   :1.240   Min.   :103.7   Min.   : 18965   Min.   :  540000  
 1st Qu.:1.309   1st Qu.:103.8   1st Qu.:259849   1st Qu.: 1100000  
 Median :1.328   Median :103.8   Median :469298   Median : 1383222  
 Mean   :1.334   Mean   :103.8   Mean   :440439   Mean   : 1751211  
 3rd Qu.:1.357   3rd Qu.:103.9   3rd Qu.:589486   3rd Qu.: 1950000  
 Max.   :1.454   Max.   :104.0   Max.   :828833   Max.   :18000000  
    AREA_SQM          AGE           PROX_CBD       PROX_CHILDCARE    
 Min.   : 34.0   Min.   : 0.00   Min.   : 0.3869   Min.   :0.004927  
 1st Qu.:103.0   1st Qu.: 5.00   1st Qu.: 5.5574   1st Qu.:0.174481  
 Median :121.0   Median :11.00   Median : 9.3567   Median :0.258135  
 Mean   :136.5   Mean   :12.14   Mean   : 9.3254   Mean   :0.326313  
 3rd Qu.:156.0   3rd Qu.:18.00   3rd Qu.:12.6661   3rd Qu.:0.368293  
 Max.   :619.0   Max.   :37.00   Max.   :19.1804   Max.   :3.465726  
 PROX_ELDERLYCARE  PROX_URA_GROWTH_AREA PROX_HAWKER_MARKET PROX_KINDERGARTEN 
 Min.   :0.05451   Min.   :0.2145       Min.   :0.05182    Min.   :0.004927  
 1st Qu.:0.61254   1st Qu.:3.1643       1st Qu.:0.55245    1st Qu.:0.276345  
 Median :0.94179   Median :4.6186       Median :0.90842    Median :0.413385  
 Mean   :1.05351   Mean   :4.5981       Mean   :1.27987    Mean   :0.458903  
 3rd Qu.:1.35122   3rd Qu.:5.7550       3rd Qu.:1.68578    3rd Qu.:0.578474  
 Max.   :3.94916   Max.   :9.1554       Max.   :5.37435    Max.   :2.229045  
    PROX_MRT         PROX_PARK       PROX_PRIMARY_SCH  PROX_TOP_PRIMARY_SCH
 Min.   :0.05278   Min.   :0.02906   Min.   :0.07711   Min.   :0.07711     
 1st Qu.:0.34646   1st Qu.:0.26211   1st Qu.:0.44024   1st Qu.:1.34451     
 Median :0.57430   Median :0.39926   Median :0.63505   Median :1.88213     
 Mean   :0.67316   Mean   :0.49802   Mean   :0.75471   Mean   :2.27347     
 3rd Qu.:0.84844   3rd Qu.:0.65592   3rd Qu.:0.95104   3rd Qu.:2.90954     
 Max.   :3.48037   Max.   :2.16105   Max.   :3.92899   Max.   :6.74819     
 PROX_SHOPPING_MALL PROX_SUPERMARKET PROX_BUS_STOP       NO_Of_UNITS    
 Min.   :0.0000     Min.   :0.0000   Min.   :0.001595   Min.   :  18.0  
 1st Qu.:0.5258     1st Qu.:0.3695   1st Qu.:0.098356   1st Qu.: 188.8  
 Median :0.9357     Median :0.5687   Median :0.151710   Median : 360.0  
 Mean   :1.0455     Mean   :0.6141   Mean   :0.193974   Mean   : 409.2  
 3rd Qu.:1.3994     3rd Qu.:0.7862   3rd Qu.:0.220466   3rd Qu.: 590.0  
 Max.   :3.4774     Max.   :2.2441   Max.   :2.476639   Max.   :1703.0  
 FAMILY_FRIENDLY     FREEHOLD      LEASEHOLD_99YR  
 Min.   :0.0000   Min.   :0.0000   Min.   :0.0000  
 1st Qu.:0.0000   1st Qu.:0.0000   1st Qu.:0.0000  
 Median :0.0000   Median :0.0000   Median :0.0000  
 Mean   :0.4868   Mean   :0.4227   Mean   :0.4882  
 3rd Qu.:1.0000   3rd Qu.:1.0000   3rd Qu.:1.0000  
 Max.   :1.0000   Max.   :1.0000   Max.   :1.0000  

The condo_resale is currently a tibble data.frame, convert to sf for easier future processing.

Show the code
condo_resale.sf <- st_as_sf(condo_resale,
                            coords = c("LONGITUDE", "LATITUDE"),
                            crs=4326) %>%
  st_transform(crs=3414)
head(condo_resale.sf)
Simple feature collection with 6 features and 21 fields
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: 22085.12 ymin: 29951.54 xmax: 41042.56 ymax: 34546.2
Projected CRS: SVY21 / Singapore TM
# A tibble: 6 × 22
  POSTCODE SELLING_PRICE AREA_SQM   AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE
     <dbl>         <dbl>    <dbl> <dbl>    <dbl>          <dbl>            <dbl>
1   118635       3000000      309    30     7.94          0.166            2.52 
2   288420       3880000      290    32     6.61          0.280            1.93 
3   267833       3325000      248    33     6.90          0.429            0.502
4   258380       4250000      127     7     4.04          0.395            1.99 
5   467169       1400000      145    28    11.8           0.119            1.12 
6   466472       1320000      139    22    10.3           0.125            0.789
# ℹ 15 more variables: PROX_URA_GROWTH_AREA <dbl>, PROX_HAWKER_MARKET <dbl>,
#   PROX_KINDERGARTEN <dbl>, PROX_MRT <dbl>, PROX_PARK <dbl>,
#   PROX_PRIMARY_SCH <dbl>, PROX_TOP_PRIMARY_SCH <dbl>,
#   PROX_SHOPPING_MALL <dbl>, PROX_SUPERMARKET <dbl>, PROX_BUS_STOP <dbl>,
#   NO_Of_UNITS <dbl>, FAMILY_FRIENDLY <dbl>, FREEHOLD <dbl>,
#   LEASEHOLD_99YR <dbl>, geometry <POINT [m]>

Exploratory Data analysis

Look at distribution of selling price

Show the code
ggplot(data=condo_resale.sf, aes(x=`SELLING_PRICE`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

It’s right-skewed, make that normal with log transformation.

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condo_resale.sf <- condo_resale.sf %>%
  mutate(`LOG_SELLING_PRICE` = log(SELLING_PRICE))

Plot again to see if the skewed data is normalised. We can see that it’s relatively less skewed.

Show the code
ggplot(data=condo_resale.sf, aes(x=`LOG_SELLING_PRICE`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

We want to look at the distribution of all the numeric continuous variables

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AREA_SQM <- ggplot(data=condo_resale.sf, aes(x= `AREA_SQM`)) + 
  geom_histogram(bins=20, color="black", fill="light blue")

AGE <- ggplot(data=condo_resale.sf, aes(x= `AGE`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_CBD <- ggplot(data=condo_resale.sf, aes(x= `PROX_CBD`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_CHILDCARE <- ggplot(data=condo_resale.sf, aes(x= `PROX_CHILDCARE`)) + 
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_ELDERLYCARE <- ggplot(data=condo_resale.sf, aes(x= `PROX_ELDERLYCARE`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_URA_GROWTH_AREA <- ggplot(data=condo_resale.sf, 
                               aes(x= `PROX_URA_GROWTH_AREA`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_HAWKER_MARKET <- ggplot(data=condo_resale.sf, aes(x= `PROX_HAWKER_MARKET`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_KINDERGARTEN <- ggplot(data=condo_resale.sf, aes(x= `PROX_KINDERGARTEN`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_MRT <- ggplot(data=condo_resale.sf, aes(x= `PROX_MRT`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_PARK <- ggplot(data=condo_resale.sf, aes(x= `PROX_PARK`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_PRIMARY_SCH <- ggplot(data=condo_resale.sf, aes(x= `PROX_PRIMARY_SCH`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_TOP_PRIMARY_SCH <- ggplot(data=condo_resale.sf, 
                               aes(x= `PROX_TOP_PRIMARY_SCH`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

ggarrange(AREA_SQM, AGE, PROX_CBD, PROX_CHILDCARE, PROX_ELDERLYCARE, 
          PROX_URA_GROWTH_AREA, PROX_HAWKER_MARKET, PROX_KINDERGARTEN, PROX_MRT,
          PROX_PARK, PROX_PRIMARY_SCH, PROX_TOP_PRIMARY_SCH,  
          ncol = 3, nrow = 4)

Look at the geospatial distribution of the condo prices

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tmap_mode("view")
tmap mode set to interactive viewing
Show the code
tmap_options(check.and.fix = TRUE)
tm_shape(mpsz)+
  tm_polygons()+
tm_shape(condo_resale.sf)+
  tm_dots(col = "SELLING_PRICE",
          alpha = 0.6,
          style="quantile")
Warning: The shape mpsz is invalid (after reprojection). See sf::st_is_valid
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tm_view(set.zoom.limits = c(11,14))
$tm_layout
$tm_layout$set.zoom.limits
[1] 11 14

$tm_layout$style
[1] NA


attr(,"class")
[1] "tm"
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tmap_mode("plot")
tmap mode set to plotting

Hedonic Pricing Modelling with linear regression

Simple Linear regression - take in only 1 predictor

Simple linear regression with selling price (y ie. outcome variable) against area (x ie independent predictor).

Show the code
condo.slr <- lm(formula=SELLING_PRICE ~ AREA_SQM, data = condo_resale.sf)
summary(condo.slr)

Call:
lm(formula = SELLING_PRICE ~ AREA_SQM, data = condo_resale.sf)

Residuals:
     Min       1Q   Median       3Q      Max 
-3695815  -391764   -87517   258900 13503875 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) -258121.1    63517.2  -4.064 5.09e-05 ***
AREA_SQM      14719.0      428.1  34.381  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 942700 on 1434 degrees of freedom
Multiple R-squared:  0.4518,    Adjusted R-squared:  0.4515 
F-statistic:  1182 on 1 and 1434 DF,  p-value: < 2.2e-16

The estimated equation is SELLING_PRICE = -258121.1 + 14719 * AREA_SQM

R-squared of 0.4518 indicates that model built is able to explain about 45.18% of the resale prices.

Since p-value is much smaller than 0.0001, we will reject the null hypothesis that mean is a good estimator of SELLING_PRICE. This will allow us to infer that simple linear regression model above is a good estimator of SELLING_PRICE.

Let’s see this in a scatter plot

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ggplot(data=condo_resale.sf,  
       aes(x=`AREA_SQM`, y=`SELLING_PRICE`)) +
  geom_point() +
  geom_smooth(method = lm)
`geom_smooth()` using formula = 'y ~ x'

Multivariate Linear Regression - take in > 1 predictors to predict price

Before we use this model, we need to ensure that the assumptions for this model type is fulfilled to ensure that result estimated is truely reflective of what is presented in the data.

We need to take note of

  • multicollinearity within predictor variables

This can be investigated through correlation plot.

Show the code
corrplot(cor(condo_resale[, 5:23]), diag = FALSE, order = "AOE",
         tl.pos = "td", tl.cex = 0.5, method = "number", type = "upper")

FREEHOLD highly correlated with LEASE_99YEAR, exclude LEASE_99YEAR

Show the code
condo.mlr <- lm(formula = SELLING_PRICE ~ AREA_SQM + AGE    + 
                  PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE +
                  PROX_URA_GROWTH_AREA + PROX_HAWKER_MARKET + PROX_KINDERGARTEN + 
                  PROX_MRT  + PROX_PARK + PROX_PRIMARY_SCH + 
                  PROX_TOP_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_SUPERMARKET + 
                  PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
                data=condo_resale.sf)
summary(condo.mlr)

Call:
lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + PROX_CHILDCARE + 
    PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_HAWKER_MARKET + 
    PROX_KINDERGARTEN + PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + 
    PROX_TOP_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_SUPERMARKET + 
    PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
    data = condo_resale.sf)

Residuals:
     Min       1Q   Median       3Q      Max 
-3475964  -293923   -23069   241043 12260381 

Coefficients:
                       Estimate Std. Error t value Pr(>|t|)    
(Intercept)           481728.40  121441.01   3.967 7.65e-05 ***
AREA_SQM               12708.32     369.59  34.385  < 2e-16 ***
AGE                   -24440.82    2763.16  -8.845  < 2e-16 ***
PROX_CBD              -78669.78    6768.97 -11.622  < 2e-16 ***
PROX_CHILDCARE       -351617.91  109467.25  -3.212  0.00135 ** 
PROX_ELDERLYCARE      171029.42   42110.51   4.061 5.14e-05 ***
PROX_URA_GROWTH_AREA   38474.53   12523.57   3.072  0.00217 ** 
PROX_HAWKER_MARKET     23746.10   29299.76   0.810  0.41782    
PROX_KINDERGARTEN     147468.99   82668.87   1.784  0.07466 .  
PROX_MRT             -314599.68   57947.44  -5.429 6.66e-08 ***
PROX_PARK             563280.50   66551.68   8.464  < 2e-16 ***
PROX_PRIMARY_SCH      180186.08   65237.95   2.762  0.00582 ** 
PROX_TOP_PRIMARY_SCH    2280.04   20410.43   0.112  0.91107    
PROX_SHOPPING_MALL   -206604.06   42840.60  -4.823 1.57e-06 ***
PROX_SUPERMARKET      -44991.80   77082.64  -0.584  0.55953    
PROX_BUS_STOP         683121.35  138353.28   4.938 8.85e-07 ***
NO_Of_UNITS             -231.18      89.03  -2.597  0.00951 ** 
FAMILY_FRIENDLY       140340.77   47020.55   2.985  0.00289 ** 
FREEHOLD              359913.01   49220.22   7.312 4.38e-13 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 755800 on 1417 degrees of freedom
Multiple R-squared:  0.6518,    Adjusted R-squared:  0.6474 
F-statistic: 147.4 on 18 and 1417 DF,  p-value: < 2.2e-16

The estimated equation is SELLING_PRICE = 481728.40 + 12708.32 * AREA_SQM - 24440.82 * AGE - 78669.78 * PROX_CBD - 351617.91 * PROX_CHILDCARE + 171029.42 * PROX_ELDERLYCARE + 38474.53 * PROX_URA_GROWTH_AREA + 23746.10 * PROX_HAWKER_MARKET + 147468.99 * PROX_KINDERGARTEN - 314599.68 * PROX_MRT + 563280.50 * PROX_PARK + 180186.08 * PROX_PRIMARY_SCH + 2280.04 * PROX_TOP_PRIMARY_SCH - 206604.06 * PROX_SHOPPING_MALL - 44991.80 * PROX_SUPERMARKET + 683121.35 * PROX_BUS_STOP - 231.18 * NO_Of_UNITS + 140340.77 * FAMILY_FRIENDLY + 359913.01 * FREEHOLD

R-squared of 0.6518 indicates that model built is able to explain about 65.18% of the resale prices.

Since p-value is much smaller than 0.0001, we will reject the null hypothesis that mean is a good estimator of SELLING_PRICE. This will allow us to infer that multivariate linear regression model above is also another good estimator of SELLING_PRICE.

Publication Quality Table

From the above, we only want to keep statistically significant predictor variables.

Show the code
condo.mlr1 <- lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + 
                   PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE +
                   PROX_URA_GROWTH_AREA + PROX_MRT  + PROX_PARK + 
                   PROX_PRIMARY_SCH + PROX_SHOPPING_MALL    + PROX_BUS_STOP + 
                   NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD,
                 data=condo_resale.sf)
ols_regress(condo.mlr1)
                                Model Summary                                 
-----------------------------------------------------------------------------
R                            0.807       RMSE                     751998.679 
R-Squared                    0.651       MSE                571471422208.592 
Adj. R-Squared               0.647       Coef. Var                    43.168 
Pred R-Squared               0.638       AIC                       42966.758 
MAE                     414819.628       SBC                       43051.072 
-----------------------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 
 AIC: Akaike Information Criteria 
 SBC: Schwarz Bayesian Criteria 

                                     ANOVA                                       
--------------------------------------------------------------------------------
                    Sum of                                                      
                   Squares          DF         Mean Square       F         Sig. 
--------------------------------------------------------------------------------
Regression    1.512586e+15          14        1.080418e+14    189.059    0.0000 
Residual      8.120609e+14        1421    571471422208.592                      
Total         2.324647e+15        1435                                          
--------------------------------------------------------------------------------

                                               Parameter Estimates                                                
-----------------------------------------------------------------------------------------------------------------
               model           Beta    Std. Error    Std. Beta       t        Sig           lower          upper 
-----------------------------------------------------------------------------------------------------------------
         (Intercept)     527633.222    108183.223                   4.877    0.000     315417.244     739849.200 
            AREA_SQM      12777.523       367.479        0.584     34.771    0.000      12056.663      13498.382 
                 AGE     -24687.739      2754.845       -0.167     -8.962    0.000     -30091.739     -19283.740 
            PROX_CBD     -77131.323      5763.125       -0.263    -13.384    0.000     -88436.469     -65826.176 
      PROX_CHILDCARE    -318472.751    107959.512       -0.084     -2.950    0.003    -530249.889    -106695.613 
    PROX_ELDERLYCARE     185575.623     39901.864        0.090      4.651    0.000     107302.737     263848.510 
PROX_URA_GROWTH_AREA      39163.254     11754.829        0.060      3.332    0.001      16104.571      62221.936 
            PROX_MRT    -294745.107     56916.367       -0.112     -5.179    0.000    -406394.234    -183095.980 
           PROX_PARK     570504.807     65507.029        0.150      8.709    0.000     442003.938     699005.677 
    PROX_PRIMARY_SCH     159856.136     60234.599        0.062      2.654    0.008      41697.849     278014.424 
  PROX_SHOPPING_MALL    -220947.251     36561.832       -0.115     -6.043    0.000    -292668.213    -149226.288 
       PROX_BUS_STOP     682482.221    134513.243        0.134      5.074    0.000     418616.359     946348.082 
         NO_Of_UNITS       -245.480        87.947       -0.053     -2.791    0.005       -418.000        -72.961 
     FAMILY_FRIENDLY     146307.576     46893.021        0.057      3.120    0.002      54320.593     238294.560 
            FREEHOLD     350599.812     48506.485        0.136      7.228    0.000     255447.802     445751.821 
-----------------------------------------------------------------------------------------------------------------

There’s another way to print this same table

Show the code
tbl_regression(condo.mlr1, intercept = TRUE)
Characteristic Beta 95% CI1 p-value
(Intercept) 527,633 315,417, 739,849 <0.001
AREA_SQM 12,778 12,057, 13,498 <0.001
AGE -24,688 -30,092, -19,284 <0.001
PROX_CBD -77,131 -88,436, -65,826 <0.001
PROX_CHILDCARE -318,473 -530,250, -106,696 0.003
PROX_ELDERLYCARE 185,576 107,303, 263,849 <0.001
PROX_URA_GROWTH_AREA 39,163 16,105, 62,222 <0.001
PROX_MRT -294,745 -406,394, -183,096 <0.001
PROX_PARK 570,505 442,004, 699,006 <0.001
PROX_PRIMARY_SCH 159,856 41,698, 278,014 0.008
PROX_SHOPPING_MALL -220,947 -292,668, -149,226 <0.001
PROX_BUS_STOP 682,482 418,616, 946,348 <0.001
NO_Of_UNITS -245 -418, -73 0.005
FAMILY_FRIENDLY 146,308 54,321, 238,295 0.002
FREEHOLD 350,600 255,448, 445,752 <0.001
1 CI = Confidence Interval

When we want to include model statistics, we use this

Show the code
tbl_regression(condo.mlr1, 
               intercept = TRUE) %>% 
  add_glance_source_note(
    label = list(sigma ~ "\U03C3"),
    include = c(r.squared, adj.r.squared, 
                AIC, statistic,
                p.value, sigma))
Characteristic Beta 95% CI1 p-value
(Intercept) 527,633 315,417, 739,849 <0.001
AREA_SQM 12,778 12,057, 13,498 <0.001
AGE -24,688 -30,092, -19,284 <0.001
PROX_CBD -77,131 -88,436, -65,826 <0.001
PROX_CHILDCARE -318,473 -530,250, -106,696 0.003
PROX_ELDERLYCARE 185,576 107,303, 263,849 <0.001
PROX_URA_GROWTH_AREA 39,163 16,105, 62,222 <0.001
PROX_MRT -294,745 -406,394, -183,096 <0.001
PROX_PARK 570,505 442,004, 699,006 <0.001
PROX_PRIMARY_SCH 159,856 41,698, 278,014 0.008
PROX_SHOPPING_MALL -220,947 -292,668, -149,226 <0.001
PROX_BUS_STOP 682,482 418,616, 946,348 <0.001
NO_Of_UNITS -245 -418, -73 0.005
FAMILY_FRIENDLY 146,308 54,321, 238,295 0.002
FREEHOLD 350,600 255,448, 445,752 <0.001
R² = 0.651; Adjusted R² = 0.647; AIC = 42,967; Statistic = 189; p-value = <0.001; σ = 755,957
1 CI = Confidence Interval

OLS regression

This is making use of another package to help build linear regression models, olsrr

Multicollinearity can be investigated using ols_vif_tol()

Since the values < 10, we can safely assume no multicollinearity exist.

Show the code
ols_vif_tol(condo.mlr1)
              Variables Tolerance      VIF
1              AREA_SQM 0.8728554 1.145665
2                   AGE 0.7071275 1.414172
3              PROX_CBD 0.6356147 1.573280
4        PROX_CHILDCARE 0.3066019 3.261559
5      PROX_ELDERLYCARE 0.6598479 1.515501
6  PROX_URA_GROWTH_AREA 0.7510311 1.331503
7              PROX_MRT 0.5236090 1.909822
8             PROX_PARK 0.8279261 1.207837
9      PROX_PRIMARY_SCH 0.4524628 2.210126
10   PROX_SHOPPING_MALL 0.6738795 1.483945
11        PROX_BUS_STOP 0.3514118 2.845664
12          NO_Of_UNITS 0.6901036 1.449058
13      FAMILY_FRIENDLY 0.7244157 1.380423
14             FREEHOLD 0.6931163 1.442759

Given that linear regression, as the name suggests, test for linear relationship between the variables and SELLING_PRICE. We also want to know if there is non-linear relationship between variables and SELLING_PRICE.

Most of the data poitns are scattered around the 0 line, hence we can safely conclude that the relationships between the dependent variable and independent variables are linear

Show the code
ols_plot_resid_fit(condo.mlr1)

One of the assumptions of linear regression is the numeric continuous variables have normal distribution.

Residual of the multiple linear regression model (i.e. condo.mlr1) is resemble normal distribution

Show the code
ols_plot_resid_hist(condo.mlr1)

This is the statistical way of testing for normality

Values of the four tests are way smaller than the alpha value of 0.05. Hence we will reject the null hypothesis and infer that there is statistical evidence that the residual are not normally distributed.

Show the code
ols_test_normality(condo.mlr1)
Warning in ks.test.default(y, "pnorm", mean(y), sd(y)): ties should not be
present for the Kolmogorov-Smirnov test
-----------------------------------------------
       Test             Statistic       pvalue  
-----------------------------------------------
Shapiro-Wilk              0.6856         0.0000 
Kolmogorov-Smirnov        0.1366         0.0000 
Cramer-von Mises         121.0768        0.0000 
Anderson-Darling         67.9551         0.0000 
-----------------------------------------------

The hedonic model we try to build are using geographically referenced attributes, hence it is also important for us to visualise the residual of the hedonic pricing model.

Show the code
 mlr.output <- as.data.frame(condo.mlr1$residuals)
condo_resale.res.sf <- cbind(condo_resale.sf, 
                        condo.mlr1$residuals) %>%
  rename(`MLR_RES` = `condo.mlr1.residuals`)
condo_resale.sp <- as_Spatial(condo_resale.res.sf)
condo_resale.sp
class       : SpatialPointsDataFrame 
features    : 1436 
extent      : 14940.85, 43352.45, 24765.67, 48382.81  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs 
variables   : 23
names       : POSTCODE, SELLING_PRICE, AREA_SQM, AGE,    PROX_CBD, PROX_CHILDCARE, PROX_ELDERLYCARE, PROX_URA_GROWTH_AREA, PROX_HAWKER_MARKET, PROX_KINDERGARTEN,    PROX_MRT,   PROX_PARK, PROX_PRIMARY_SCH, PROX_TOP_PRIMARY_SCH, PROX_SHOPPING_MALL, ... 
min values  :    18965,        540000,       34,   0, 0.386916393,    0.004927023,      0.054508623,          0.214539508,        0.051817113,       0.004927023, 0.052779424, 0.029064164,      0.077106132,          0.077106132,                  0, ... 
max values  :   828833,       1.8e+07,      619,  37, 19.18042832,     3.46572633,      3.949157205,           9.15540001,        5.374348075,       2.229045366,  3.48037319,  2.16104919,      3.928989144,          6.748192062,        3.477433767, ... 

Visualising using tmap

Graph show signs of spatial autocorrelation.

Show the code
tmap_mode("view")
tmap mode set to interactive viewing
Show the code
tm_shape(mpsz)+
  tmap_options(check.and.fix = TRUE) +
  tm_polygons(alpha = 0.4) +
tm_shape(condo_resale.res.sf) +  
  tm_dots(col = "MLR_RES",
          alpha = 0.6,
          style="quantile") +
  tm_view(set.zoom.limits = c(11,14))
Warning: The shape mpsz is invalid (after reprojection). See sf::st_is_valid
Variable(s) "MLR_RES" contains positive and negative values, so midpoint is set to 0. Set midpoint = NA to show the full spectrum of the color palette.
Show the code
tmap_mode("plot")
tmap mode set to plotting

Geospatial Autocorrelation

Since we see signs, time to handle.

Show the code
nb <- dnearneigh(coordinates(condo_resale.sp), 0, 1500, longlat = FALSE)
summary(nb)
Neighbour list object:
Number of regions: 1436 
Number of nonzero links: 66266 
Percentage nonzero weights: 3.213526 
Average number of links: 46.14624 
10 disjoint connected subgraphs
Link number distribution:

  1   3   5   7   9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24 
  3   3   9   4   3  15  10  19  17  45  19   5  14  29  19   6  35  45  18  47 
 25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44 
 16  43  22  26  21  11   9  23  22  13  16  25  21  37  16  18   8  21   4  12 
 45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 
  8  36  18  14  14  43  11  12   8  13  12  13   4   5   6  12  11  20  29  33 
 65  66  67  68  69  70  71  72  73  74  75  76  77  78  79  80  81  82  83  84 
 15  20  10  14  15  15  11  16  12  10   8  19  12  14   9   8   4  13  11   6 
 85  86  87  88  89  90  91  92  93  94  95  96  97  98  99 100 101 102 103 104 
  4   9   4   4   4   6   2  16   9   4   5   9   3   9   4   2   1   2   1   1 
105 106 107 108 109 110 112 116 125 
  1   5   9   2   1   3   1   1   1 
3 least connected regions:
193 194 277 with 1 link
1 most connected region:
285 with 125 links
Show the code
nb_lw <- nb2listw(nb, style = 'W')
summary(nb_lw)
Characteristics of weights list object:
Neighbour list object:
Number of regions: 1436 
Number of nonzero links: 66266 
Percentage nonzero weights: 3.213526 
Average number of links: 46.14624 
10 disjoint connected subgraphs
Link number distribution:

  1   3   5   7   9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24 
  3   3   9   4   3  15  10  19  17  45  19   5  14  29  19   6  35  45  18  47 
 25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44 
 16  43  22  26  21  11   9  23  22  13  16  25  21  37  16  18   8  21   4  12 
 45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 
  8  36  18  14  14  43  11  12   8  13  12  13   4   5   6  12  11  20  29  33 
 65  66  67  68  69  70  71  72  73  74  75  76  77  78  79  80  81  82  83  84 
 15  20  10  14  15  15  11  16  12  10   8  19  12  14   9   8   4  13  11   6 
 85  86  87  88  89  90  91  92  93  94  95  96  97  98  99 100 101 102 103 104 
  4   9   4   4   4   6   2  16   9   4   5   9   3   9   4   2   1   2   1   1 
105 106 107 108 109 110 112 116 125 
  1   5   9   2   1   3   1   1   1 
3 least connected regions:
193 194 277 with 1 link
1 most connected region:
285 with 125 links

Weights style: W 
Weights constants summary:
     n      nn   S0       S1       S2
W 1436 2062096 1436 94.81916 5798.341
Show the code
lm.morantest(condo.mlr1, nb_lw)

    Global Moran I for regression residuals

data:  
model: lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD +
PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_MRT +
PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_BUS_STOP +
NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, data = condo_resale.sf)
weights: nb_lw

Moran I statistic standard deviate = 24.366, p-value < 2.2e-16
alternative hypothesis: greater
sample estimates:
Observed Moran I      Expectation         Variance 
    1.438876e-01    -5.487594e-03     3.758259e-05 

The Global Moran’s I test for residual spatial autocorrelation shows that it’s p-value is less than 0.00000000000000022 which is less than the alpha value of 0.05. Hence, we will reject the null hypothesis that the residuals are randomly distributed.

Since the Observed Global Moran I = 0.1424418 which is greater than 0, we can infer than the residuals resemble cluster distribution.

Hedonic Pricing model with olsrr

Fixed bandwidth GWR

Show the code
bw.fixed <- bw.gwr(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + 
                     PROX_CHILDCARE + PROX_ELDERLYCARE  + PROX_URA_GROWTH_AREA + 
                     PROX_MRT   + PROX_PARK + PROX_PRIMARY_SCH + 
                     PROX_SHOPPING_MALL + PROX_BUS_STOP + NO_Of_UNITS + 
                     FAMILY_FRIENDLY + FREEHOLD, 
                   data=condo_resale.sp, 
                   approach="CV", 
                   kernel="gaussian", 
                   adaptive=FALSE, 
                   longlat=FALSE)
Fixed bandwidth: 17660.96 CV score: 8.259118e+14 
Fixed bandwidth: 10917.26 CV score: 7.970454e+14 
Fixed bandwidth: 6749.419 CV score: 7.273273e+14 
Fixed bandwidth: 4173.553 CV score: 6.300006e+14 
Fixed bandwidth: 2581.58 CV score: 5.404958e+14 
Fixed bandwidth: 1597.687 CV score: 4.857515e+14 
Fixed bandwidth: 989.6077 CV score: 4.722431e+14 
Fixed bandwidth: 613.7939 CV score: 1.379526e+16 
Fixed bandwidth: 1221.873 CV score: 4.778717e+14 
Fixed bandwidth: 846.0596 CV score: 4.791629e+14 
Fixed bandwidth: 1078.325 CV score: 4.751406e+14 
Fixed bandwidth: 934.7772 CV score: 4.72518e+14 
Fixed bandwidth: 1023.495 CV score: 4.730305e+14 
Fixed bandwidth: 968.6643 CV score: 4.721317e+14 
Fixed bandwidth: 955.7206 CV score: 4.722072e+14 
Fixed bandwidth: 976.6639 CV score: 4.721387e+14 
Fixed bandwidth: 963.7202 CV score: 4.721484e+14 
Fixed bandwidth: 971.7199 CV score: 4.721293e+14 
Fixed bandwidth: 973.6083 CV score: 4.721309e+14 
Fixed bandwidth: 970.5527 CV score: 4.721295e+14 
Fixed bandwidth: 972.4412 CV score: 4.721296e+14 
Fixed bandwidth: 971.2741 CV score: 4.721292e+14 
Fixed bandwidth: 970.9985 CV score: 4.721293e+14 
Fixed bandwidth: 971.4443 CV score: 4.721292e+14 
Fixed bandwidth: 971.5496 CV score: 4.721293e+14 
Fixed bandwidth: 971.3793 CV score: 4.721292e+14 
Fixed bandwidth: 971.3391 CV score: 4.721292e+14 
Fixed bandwidth: 971.3143 CV score: 4.721292e+14 
Fixed bandwidth: 971.3545 CV score: 4.721292e+14 
Fixed bandwidth: 971.3296 CV score: 4.721292e+14 
Fixed bandwidth: 971.345 CV score: 4.721292e+14 
Fixed bandwidth: 971.3355 CV score: 4.721292e+14 
Fixed bandwidth: 971.3413 CV score: 4.721292e+14 
Fixed bandwidth: 971.3377 CV score: 4.721292e+14 
Fixed bandwidth: 971.34 CV score: 4.721292e+14 
Fixed bandwidth: 971.3405 CV score: 4.721292e+14 
Fixed bandwidth: 971.3396 CV score: 4.721292e+14 
Fixed bandwidth: 971.3402 CV score: 4.721292e+14 
Fixed bandwidth: 971.3398 CV score: 4.721292e+14 
Fixed bandwidth: 971.34 CV score: 4.721292e+14 
Fixed bandwidth: 971.3399 CV score: 4.721292e+14 
Fixed bandwidth: 971.34 CV score: 4.721292e+14 

Recommended bandwidth: 971.34

Show the code
gwr.fixed <- gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + 
                         PROX_CHILDCARE + PROX_ELDERLYCARE  + PROX_URA_GROWTH_AREA + 
                         PROX_MRT   + PROX_PARK + PROX_PRIMARY_SCH + 
                         PROX_SHOPPING_MALL + PROX_BUS_STOP + NO_Of_UNITS + 
                         FAMILY_FRIENDLY + FREEHOLD, 
                       data=condo_resale.sp, 
                       bw=bw.fixed, 
                       kernel = 'gaussian', 
                       longlat = FALSE)
gwr.fixed
   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2024-03-17 22:25:30.500389 
   Call:
   gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + 
    PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + 
    PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + 
    PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
    data = condo_resale.sp, bw = bw.fixed, kernel = "gaussian", 
    longlat = FALSE)

   Dependent (y) variable:  SELLING_PRICE
   Independent variables:  AREA_SQM AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE PROX_URA_GROWTH_AREA PROX_MRT PROX_PARK PROX_PRIMARY_SCH PROX_SHOPPING_MALL PROX_BUS_STOP NO_Of_UNITS FAMILY_FRIENDLY FREEHOLD
   Number of data points: 1436
   ***********************************************************************
   *                    Results of Global Regression                     *
   ***********************************************************************

   Call:
    lm(formula = formula, data = data)

   Residuals:
     Min       1Q   Median       3Q      Max 
-3470778  -298119   -23481   248917 12234210 

   Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
   (Intercept)           527633.22  108183.22   4.877 1.20e-06 ***
   AREA_SQM               12777.52     367.48  34.771  < 2e-16 ***
   AGE                   -24687.74    2754.84  -8.962  < 2e-16 ***
   PROX_CBD              -77131.32    5763.12 -13.384  < 2e-16 ***
   PROX_CHILDCARE       -318472.75  107959.51  -2.950 0.003231 ** 
   PROX_ELDERLYCARE      185575.62   39901.86   4.651 3.61e-06 ***
   PROX_URA_GROWTH_AREA   39163.25   11754.83   3.332 0.000885 ***
   PROX_MRT             -294745.11   56916.37  -5.179 2.56e-07 ***
   PROX_PARK             570504.81   65507.03   8.709  < 2e-16 ***
   PROX_PRIMARY_SCH      159856.14   60234.60   2.654 0.008046 ** 
   PROX_SHOPPING_MALL   -220947.25   36561.83  -6.043 1.93e-09 ***
   PROX_BUS_STOP         682482.22  134513.24   5.074 4.42e-07 ***
   NO_Of_UNITS             -245.48      87.95  -2.791 0.005321 ** 
   FAMILY_FRIENDLY       146307.58   46893.02   3.120 0.001845 ** 
   FREEHOLD              350599.81   48506.48   7.228 7.98e-13 ***

   ---Significance stars
   Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
   Residual standard error: 756000 on 1421 degrees of freedom
   Multiple R-squared: 0.6507
   Adjusted R-squared: 0.6472 
   F-statistic: 189.1 on 14 and 1421 DF,  p-value: < 2.2e-16 
   ***Extra Diagnostic information
   Residual sum of squares: 8.120609e+14
   Sigma(hat): 752522.9
   AIC:  42966.76
   AICc:  42967.14
   BIC:  41731.39
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian 
   Fixed bandwidth: 971.34 
   Regression points: the same locations as observations are used.
   Distance metric: Euclidean distance metric is used.

   ****************Summary of GWR coefficient estimates:******************
                               Min.     1st Qu.      Median     3rd Qu.
   Intercept            -3.5988e+07 -5.1998e+05  7.6780e+05  1.7412e+06
   AREA_SQM              1.0003e+03  5.2758e+03  7.4740e+03  1.2301e+04
   AGE                  -1.3475e+05 -2.0813e+04 -8.6260e+03 -3.7784e+03
   PROX_CBD             -7.7047e+07 -2.3608e+05 -8.3599e+04  3.4646e+04
   PROX_CHILDCARE       -6.0097e+06 -3.3667e+05 -9.7426e+04  2.9007e+05
   PROX_ELDERLYCARE     -3.5001e+06 -1.5970e+05  3.1970e+04  1.9577e+05
   PROX_URA_GROWTH_AREA -3.0170e+06 -8.2013e+04  7.0749e+04  2.2612e+05
   PROX_MRT             -3.5282e+06 -6.5836e+05 -1.8833e+05  3.6922e+04
   PROX_PARK            -1.2062e+06 -2.1732e+05  3.5383e+04  4.1335e+05
   PROX_PRIMARY_SCH     -2.2695e+07 -1.7066e+05  4.8472e+04  5.1555e+05
   PROX_SHOPPING_MALL   -7.2585e+06 -1.6684e+05 -1.0517e+04  1.5923e+05
   PROX_BUS_STOP        -1.4676e+06 -4.5207e+04  3.7601e+05  1.1664e+06
   NO_Of_UNITS          -1.3170e+03 -2.4822e+02 -3.0846e+01  2.5496e+02
   FAMILY_FRIENDLY      -2.2749e+06 -1.1140e+05  7.6214e+03  1.6107e+05
   FREEHOLD             -9.2067e+06  3.8074e+04  1.5169e+05  3.7528e+05
                             Max.
   Intercept            112794435
   AREA_SQM                 21575
   AGE                     434203
   PROX_CBD               2704604
   PROX_CHILDCARE         1654086
   PROX_ELDERLYCARE      38867861
   PROX_URA_GROWTH_AREA  78515805
   PROX_MRT               3124325
   PROX_PARK             18122439
   PROX_PRIMARY_SCH       4637517
   PROX_SHOPPING_MALL     1529953
   PROX_BUS_STOP         11342209
   NO_Of_UNITS              12907
   FAMILY_FRIENDLY        1720745
   FREEHOLD               6073642
   ************************Diagnostic information*************************
   Number of data points: 1436 
   Effective number of parameters (2trace(S) - trace(S'S)): 438.3807 
   Effective degrees of freedom (n-2trace(S) + trace(S'S)): 997.6193 
   AICc (GWR book, Fotheringham, et al. 2002, p. 61, eq 2.33): 42263.61 
   AIC (GWR book, Fotheringham, et al. 2002,GWR p. 96, eq. 4.22): 41632.36 
   BIC (GWR book, Fotheringham, et al. 2002,GWR p. 61, eq. 2.34): 42515.71 
   Residual sum of squares: 2.534069e+14 
   R-square value:  0.8909912 
   Adjusted R-square value:  0.8430418 

   ***********************************************************************
   Program stops at: 2024-03-17 22:25:31.541227 

AICc of the gwr is 42263.61 which is significantly smaller than the globel multiple linear regression model of 42967.1

Adaptive bandwidth GWR

Show the code
bw.adaptive <- bw.gwr(formula = SELLING_PRICE ~ AREA_SQM + AGE  + 
                        PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE    + 
                        PROX_URA_GROWTH_AREA + PROX_MRT + PROX_PARK + 
                        PROX_PRIMARY_SCH + PROX_SHOPPING_MALL   + PROX_BUS_STOP + 
                        NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
                      data=condo_resale.sp, 
                      approach="CV", 
                      kernel="gaussian", 
                      adaptive=TRUE, 
                      longlat=FALSE)
Adaptive bandwidth: 895 CV score: 7.952401e+14 
Adaptive bandwidth: 561 CV score: 7.667364e+14 
Adaptive bandwidth: 354 CV score: 6.953454e+14 
Adaptive bandwidth: 226 CV score: 6.15223e+14 
Adaptive bandwidth: 147 CV score: 5.674373e+14 
Adaptive bandwidth: 98 CV score: 5.426745e+14 
Adaptive bandwidth: 68 CV score: 5.168117e+14 
Adaptive bandwidth: 49 CV score: 4.859631e+14 
Adaptive bandwidth: 37 CV score: 4.646518e+14 
Adaptive bandwidth: 30 CV score: 4.422088e+14 
Adaptive bandwidth: 25 CV score: 4.430816e+14 
Adaptive bandwidth: 32 CV score: 4.505602e+14 
Adaptive bandwidth: 27 CV score: 4.462172e+14 
Adaptive bandwidth: 30 CV score: 4.422088e+14 

Recommended number of datapoints in each bandwidth: 30

Show the code
gwr.adaptive <- gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + 
                            PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE + 
                            PROX_URA_GROWTH_AREA + PROX_MRT + PROX_PARK + 
                            PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_BUS_STOP + 
                            NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
                          data=condo_resale.sp, bw=bw.adaptive, 
                          kernel = 'gaussian', 
                          adaptive=TRUE, 
                          longlat = FALSE)
gwr.adaptive
   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2024-03-17 22:25:39.174313 
   Call:
   gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + 
    PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + 
    PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + 
    PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
    data = condo_resale.sp, bw = bw.adaptive, kernel = "gaussian", 
    adaptive = TRUE, longlat = FALSE)

   Dependent (y) variable:  SELLING_PRICE
   Independent variables:  AREA_SQM AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE PROX_URA_GROWTH_AREA PROX_MRT PROX_PARK PROX_PRIMARY_SCH PROX_SHOPPING_MALL PROX_BUS_STOP NO_Of_UNITS FAMILY_FRIENDLY FREEHOLD
   Number of data points: 1436
   ***********************************************************************
   *                    Results of Global Regression                     *
   ***********************************************************************

   Call:
    lm(formula = formula, data = data)

   Residuals:
     Min       1Q   Median       3Q      Max 
-3470778  -298119   -23481   248917 12234210 

   Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
   (Intercept)           527633.22  108183.22   4.877 1.20e-06 ***
   AREA_SQM               12777.52     367.48  34.771  < 2e-16 ***
   AGE                   -24687.74    2754.84  -8.962  < 2e-16 ***
   PROX_CBD              -77131.32    5763.12 -13.384  < 2e-16 ***
   PROX_CHILDCARE       -318472.75  107959.51  -2.950 0.003231 ** 
   PROX_ELDERLYCARE      185575.62   39901.86   4.651 3.61e-06 ***
   PROX_URA_GROWTH_AREA   39163.25   11754.83   3.332 0.000885 ***
   PROX_MRT             -294745.11   56916.37  -5.179 2.56e-07 ***
   PROX_PARK             570504.81   65507.03   8.709  < 2e-16 ***
   PROX_PRIMARY_SCH      159856.14   60234.60   2.654 0.008046 ** 
   PROX_SHOPPING_MALL   -220947.25   36561.83  -6.043 1.93e-09 ***
   PROX_BUS_STOP         682482.22  134513.24   5.074 4.42e-07 ***
   NO_Of_UNITS             -245.48      87.95  -2.791 0.005321 ** 
   FAMILY_FRIENDLY       146307.58   46893.02   3.120 0.001845 ** 
   FREEHOLD              350599.81   48506.48   7.228 7.98e-13 ***

   ---Significance stars
   Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
   Residual standard error: 756000 on 1421 degrees of freedom
   Multiple R-squared: 0.6507
   Adjusted R-squared: 0.6472 
   F-statistic: 189.1 on 14 and 1421 DF,  p-value: < 2.2e-16 
   ***Extra Diagnostic information
   Residual sum of squares: 8.120609e+14
   Sigma(hat): 752522.9
   AIC:  42966.76
   AICc:  42967.14
   BIC:  41731.39
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian 
   Adaptive bandwidth: 30 (number of nearest neighbours)
   Regression points: the same locations as observations are used.
   Distance metric: Euclidean distance metric is used.

   ****************Summary of GWR coefficient estimates:******************
                               Min.     1st Qu.      Median     3rd Qu.
   Intercept            -1.3487e+08 -2.4669e+05  7.7928e+05  1.6194e+06
   AREA_SQM              3.3188e+03  5.6285e+03  7.7825e+03  1.2738e+04
   AGE                  -9.6746e+04 -2.9288e+04 -1.4043e+04 -5.6119e+03
   PROX_CBD             -2.5330e+06 -1.6256e+05 -7.7242e+04  2.6624e+03
   PROX_CHILDCARE       -1.2790e+06 -2.0175e+05  8.7158e+03  3.7778e+05
   PROX_ELDERLYCARE     -1.6212e+06 -9.2050e+04  6.1029e+04  2.8184e+05
   PROX_URA_GROWTH_AREA -7.2686e+06 -3.0350e+04  4.5869e+04  2.4613e+05
   PROX_MRT             -4.3781e+07 -6.7282e+05 -2.2115e+05 -7.4593e+04
   PROX_PARK            -2.9020e+06 -1.6782e+05  1.1601e+05  4.6572e+05
   PROX_PRIMARY_SCH     -8.6418e+05 -1.6627e+05 -7.7853e+03  4.3222e+05
   PROX_SHOPPING_MALL   -1.8272e+06 -1.3175e+05 -1.4049e+04  1.3799e+05
   PROX_BUS_STOP        -2.0579e+06 -7.1461e+04  4.1104e+05  1.2071e+06
   NO_Of_UNITS          -2.1993e+03 -2.3685e+02 -3.4699e+01  1.1657e+02
   FAMILY_FRIENDLY      -5.9879e+05 -5.0927e+04  2.6173e+04  2.2481e+05
   FREEHOLD             -1.6340e+05  4.0765e+04  1.9023e+05  3.7960e+05
                            Max.
   Intercept            18758355
   AREA_SQM                23064
   AGE                     13303
   PROX_CBD             11346650
   PROX_CHILDCARE        2892127
   PROX_ELDERLYCARE      2465671
   PROX_URA_GROWTH_AREA  7384059
   PROX_MRT              1186242
   PROX_PARK             2588497
   PROX_PRIMARY_SCH      3381462
   PROX_SHOPPING_MALL   38038564
   PROX_BUS_STOP        12081592
   NO_Of_UNITS              1010
   FAMILY_FRIENDLY       2072414
   FREEHOLD              1813995
   ************************Diagnostic information*************************
   Number of data points: 1436 
   Effective number of parameters (2trace(S) - trace(S'S)): 350.3088 
   Effective degrees of freedom (n-2trace(S) + trace(S'S)): 1085.691 
   AICc (GWR book, Fotheringham, et al. 2002, p. 61, eq 2.33): 41982.22 
   AIC (GWR book, Fotheringham, et al. 2002,GWR p. 96, eq. 4.22): 41546.74 
   BIC (GWR book, Fotheringham, et al. 2002,GWR p. 61, eq. 2.34): 41914.08 
   Residual sum of squares: 2.528227e+14 
   R-square value:  0.8912425 
   Adjusted R-square value:  0.8561185 

   ***********************************************************************
   Program stops at: 2024-03-17 22:25:40.275582 

AICc the adaptive distance gwr is 41982.22 which is even smaller than the AICc of the fixed distance gwr of 42263.61.

Visualising GWR output

Show the code
condo_resale.sf.adaptive <- st_as_sf(gwr.adaptive$SDF) %>%
  st_transform(crs=3414)
condo_resale.sf.adaptive.svy21 <- st_transform(condo_resale.sf.adaptive, 3414)
condo_resale.sf.adaptive.svy21
Simple feature collection with 1436 features and 51 fields
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: 14940.85 ymin: 24765.67 xmax: 43352.45 ymax: 48382.81
Projected CRS: SVY21 / Singapore TM
First 10 features:
    Intercept  AREA_SQM        AGE  PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE
1   2050011.7  9561.892  -9514.634 -120681.9      319266.92       -393417.79
2   1633128.2 16576.853 -58185.479 -149434.2      441102.18        325188.74
3   3433608.2 13091.861 -26707.386 -259397.8     -120116.82        535855.81
4    234358.9 20730.601 -93308.988 2426853.7      480825.28        314783.72
5   2285804.9  6722.836 -17608.018 -316835.5       90764.78       -137384.61
6  -3568877.4  6039.581 -26535.592  327306.1     -152531.19       -700392.85
7  -2874842.4 16843.575 -59166.727 -983577.2     -177810.50       -122384.02
8   2038086.0  6905.135 -17681.897 -285076.6       70259.40        -96012.78
9   1718478.4  9580.703 -14401.128  105803.4     -657698.02       -123276.00
10  3457054.0 14072.011 -31579.884 -234895.4       79961.45        548581.04
   PROX_URA_GROWTH_AREA    PROX_MRT  PROX_PARK PROX_PRIMARY_SCH
1            -159980.20  -299742.96 -172104.47        242668.03
2            -142290.39 -2510522.23  523379.72       1106830.66
3            -253621.21  -936853.28  209099.85        571462.33
4           -2679297.89 -2039479.50 -759153.26       3127477.21
5             303714.81   -44567.05  -10284.62         30413.56
6             -28051.25   733566.47 1511488.92        320878.23
7            1397676.38 -2745430.34  710114.74       1786570.95
8             269368.71   -14552.99   73533.34         53359.73
9            -361974.72  -476785.32 -132067.59        -40128.92
10           -150024.38 -1503835.53  574155.47        108996.67
   PROX_SHOPPING_MALL PROX_BUS_STOP  NO_Of_UNITS FAMILY_FRIENDLY  FREEHOLD
1          300881.390     1210615.4  104.8290640       -9075.370  303955.6
2          -87693.378     1843587.2 -288.3441183      310074.664  396221.3
3         -126732.712     1411924.9   -9.5532945        5949.746  168821.7
4          -29593.342     7225577.5 -161.3551620     1556178.531 1212515.6
5           -7490.586      677577.0   42.2659674       58986.951  328175.2
6          258583.881     1086012.6 -214.3671271      201992.641  471873.1
7         -384251.210     5094060.5   -0.9212521      359659.512  408871.9
8          -39634.902      735767.1   30.1741069       55602.506  347075.0
9          276718.757     2815772.4  675.1615559      -30453.297  503872.8
10        -454726.822     2123557.0  -21.3044311     -100935.586  213324.6
         y    yhat    residual CV_Score Stud_residual Intercept_SE AREA_SQM_SE
1  3000000 2886532   113468.16        0    0.38207013     516105.5    823.2860
2  3880000 3466801   413198.52        0    1.01433140     488083.5    825.2380
3  3325000 3616527  -291527.20        0   -0.83780678     963711.4    988.2240
4  4250000 5435482 -1185481.63        0   -2.84614670     444185.5    617.4007
5  1400000 1388166    11834.26        0    0.03404453    2119620.6   1376.2778
6  1320000 1516702  -196701.95        0   -0.72065801   28572883.7   2348.0091
7  3410000 3266881   143118.77        0    0.41291992     679546.6    893.5893
8  1420000 1431955   -11955.27        0   -0.03033109    2217773.1   1415.2604
9  2025000 1832799   192200.83        0    0.52018109     814281.8    943.8434
10 2550000 2223364   326635.53        0    1.10559735    2410252.0   1271.4073
      AGE_SE PROX_CBD_SE PROX_CHILDCARE_SE PROX_ELDERLYCARE_SE
1   5889.782    37411.22          319111.1           120633.34
2   6226.916    23615.06          299705.3            84546.69
3   6510.236    56103.77          349128.5           129687.07
4   6010.511   469337.41          304965.2           127150.69
5   8180.361   410644.47          698720.6           327371.55
6  14601.909  5272846.47         1141599.8          1653002.19
7   8970.629   346164.20          530101.1           148598.71
8   8661.309   438035.69          742532.8           399221.05
9  11791.208    89148.35          704630.7           329683.30
10  9941.980   173532.77          500976.2           281876.74
   PROX_URA_GROWTH_AREA_SE PROX_MRT_SE PROX_PARK_SE PROX_PRIMARY_SCH_SE
1                 56207.39    185181.3     205499.6            152400.7
2                 76956.50    281133.9     229358.7            165150.7
3                 95774.60    275483.7     314124.3            196662.6
4                470762.12    279877.1     227249.4            240878.9
5                474339.56    363830.0     364580.9            249087.7
6               5496627.21    730453.2    1741712.0            683265.5
7                371692.97    375511.9     297400.9            344602.8
8                517977.91    423155.4     440984.4            261251.2
9                153436.22    285325.4     304998.4            278258.5
10               239182.57    571355.7     599131.8            331284.8
   PROX_SHOPPING_MALL_SE PROX_BUS_STOP_SE NO_Of_UNITS_SE FAMILY_FRIENDLY_SE
1               109268.8         600668.6       218.1258           131474.7
2                98906.8         410222.1       208.9410           114989.1
3               119913.3         464156.7       210.9828           146607.2
4               177104.1         562810.8       361.7767           108726.6
5               301032.9         740922.4       299.5034           160663.7
6              2931208.6        1418333.3       602.5571           331727.0
7               249969.5         821236.4       532.1978           129241.2
8               351634.0         775038.4       338.6777           171895.1
9               289872.7         850095.5       439.9037           220223.4
10              265529.7         631399.2       259.0169           189125.5
   FREEHOLD_SE Intercept_TV AREA_SQM_TV     AGE_TV PROX_CBD_TV
1     115954.0    3.9720784   11.614302  -1.615447 -3.22582173
2     130110.0    3.3460017   20.087361  -9.344188 -6.32792021
3     141031.5    3.5629010   13.247868  -4.102368 -4.62353528
4     138239.1    0.5276150   33.577223 -15.524302  5.17080808
5     210641.1    1.0784029    4.884795  -2.152474 -0.77155660
6     374347.3   -0.1249043    2.572214  -1.817269  0.06207388
7     182216.9   -4.2305303   18.849348  -6.595605 -2.84136028
8     216649.4    0.9189786    4.879056  -2.041481 -0.65080678
9     220473.7    2.1104224   10.150733  -1.221345  1.18682383
10    206346.2    1.4343123   11.068059  -3.176418 -1.35360852
   PROX_CHILDCARE_TV PROX_ELDERLYCARE_TV PROX_URA_GROWTH_AREA_TV PROX_MRT_TV
1         1.00048819          -3.2612693            -2.846248368 -1.61864578
2         1.47178634           3.8462625            -1.848971738 -8.92998600
3        -0.34404755           4.1319138            -2.648105057 -3.40075727
4         1.57665606           2.4756745            -5.691404992 -7.28705261
5         0.12990138          -0.4196596             0.640289855 -0.12249416
6        -0.13361179          -0.4237096            -0.005103357  1.00426206
7        -0.33542751          -0.8235874             3.760298131 -7.31116712
8         0.09462126          -0.2405003             0.520038994 -0.03439159
9        -0.93339393          -0.3739225            -2.359121712 -1.67102293
10        0.15961128           1.9461735            -0.627237944 -2.63204802
   PROX_PARK_TV PROX_PRIMARY_SCH_TV PROX_SHOPPING_MALL_TV PROX_BUS_STOP_TV
1   -0.83749312           1.5923022            2.75358842        2.0154464
2    2.28192684           6.7019454           -0.88662640        4.4941192
3    0.66565951           2.9058009           -1.05686949        3.0419145
4   -3.34061770          12.9836105           -0.16709578       12.8383775
5   -0.02820944           0.1220998           -0.02488294        0.9145046
6    0.86781794           0.4696245            0.08821750        0.7656963
7    2.38773567           5.1844351           -1.53719231        6.2029165
8    0.16674816           0.2042469           -0.11271635        0.9493299
9   -0.43301073          -0.1442145            0.95462153        3.3123012
10   0.95831249           0.3290120           -1.71252687        3.3632555
   NO_Of_UNITS_TV FAMILY_FRIENDLY_TV FREEHOLD_TV  Local_R2
1     0.480589953        -0.06902748    2.621347 0.8846744
2    -1.380026395         2.69655779    3.045280 0.8899773
3    -0.045279967         0.04058290    1.197050 0.8947007
4    -0.446007570        14.31276425    8.771149 0.9073605
5     0.141120178         0.36714544    1.557983 0.9510057
6    -0.355762335         0.60891234    1.260522 0.9247586
7    -0.001731033         2.78285441    2.243875 0.8310458
8     0.089093858         0.32346758    1.602012 0.9463936
9     1.534793921        -0.13828365    2.285410 0.8380365
10   -0.082251138        -0.53369623    1.033819 0.9080753
                    geometry
1  POINT (22085.12 29951.54)
2   POINT (25656.84 34546.2)
3   POINT (23963.99 32890.8)
4  POINT (27044.28 32319.77)
5  POINT (41042.56 33743.64)
6   POINT (39717.04 32943.1)
7   POINT (28419.1 33513.37)
8  POINT (40763.57 33879.61)
9  POINT (23595.63 28884.78)
10 POINT (24586.56 33194.31)
Show the code
gwr.adaptive.output <- as.data.frame(gwr.adaptive$SDF)
condo_resale.sf.adaptive <- cbind(condo_resale.res.sf, as.matrix(gwr.adaptive.output))
glimpse(condo_resale.sf.adaptive)
Rows: 1,436
Columns: 77
$ POSTCODE                <dbl> 118635, 288420, 267833, 258380, 467169, 466472…
$ SELLING_PRICE           <dbl> 3000000, 3880000, 3325000, 4250000, 1400000, 1…
$ AREA_SQM                <dbl> 309, 290, 248, 127, 145, 139, 218, 141, 165, 1…
$ AGE                     <dbl> 30, 32, 33, 7, 28, 22, 24, 24, 27, 31, 17, 22,…
$ PROX_CBD                <dbl> 7.941259, 6.609797, 6.898000, 4.038861, 11.783…
$ PROX_CHILDCARE          <dbl> 0.16597932, 0.28027246, 0.42922669, 0.39473543…
$ PROX_ELDERLYCARE        <dbl> 2.5198118, 1.9333338, 0.5021395, 1.9910316, 1.…
$ PROX_URA_GROWTH_AREA    <dbl> 6.618741, 7.505109, 6.463887, 4.906512, 6.4106…
$ PROX_HAWKER_MARKET      <dbl> 1.76542207, 0.54507614, 0.37789301, 1.68259969…
$ PROX_KINDERGARTEN       <dbl> 0.05835552, 0.61592412, 0.14120309, 0.38200076…
$ PROX_MRT                <dbl> 0.5607188, 0.6584461, 0.3053433, 0.6910183, 0.…
$ PROX_PARK               <dbl> 1.1710446, 0.1992269, 0.2779886, 0.9832843, 0.…
$ PROX_PRIMARY_SCH        <dbl> 1.6340256, 0.9747834, 1.4715016, 1.4546324, 0.…
$ PROX_TOP_PRIMARY_SCH    <dbl> 3.3273195, 0.9747834, 1.4715016, 2.3006394, 0.…
$ PROX_SHOPPING_MALL      <dbl> 2.2102717, 2.9374279, 1.2256850, 0.3525671, 1.…
$ PROX_SUPERMARKET        <dbl> 0.9103958, 0.5900617, 0.4135583, 0.4162219, 0.…
$ PROX_BUS_STOP           <dbl> 0.10336166, 0.28673408, 0.28504777, 0.29872340…
$ NO_Of_UNITS             <dbl> 18, 20, 27, 30, 30, 31, 32, 32, 32, 32, 34, 34…
$ FAMILY_FRIENDLY         <dbl> 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0…
$ FREEHOLD                <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1…
$ LEASEHOLD_99YR          <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ LOG_SELLING_PRICE       <dbl> 14.91412, 15.17135, 15.01698, 15.26243, 14.151…
$ MLR_RES                 <dbl> -1489099.55, 415494.57, 194129.69, 1088992.71,…
$ Intercept               <dbl> 2050011.67, 1633128.24, 3433608.17, 234358.91,…
$ AREA_SQM.1              <dbl> 9561.892, 16576.853, 13091.861, 20730.601, 672…
$ AGE.1                   <dbl> -9514.634, -58185.479, -26707.386, -93308.988,…
$ PROX_CBD.1              <dbl> -120681.94, -149434.22, -259397.77, 2426853.66…
$ PROX_CHILDCARE.1        <dbl> 319266.925, 441102.177, -120116.816, 480825.28…
$ PROX_ELDERLYCARE.1      <dbl> -393417.795, 325188.741, 535855.806, 314783.72…
$ PROX_URA_GROWTH_AREA.1  <dbl> -159980.203, -142290.389, -253621.206, -267929…
$ PROX_MRT.1              <dbl> -299742.96, -2510522.23, -936853.28, -2039479.…
$ PROX_PARK.1             <dbl> -172104.47, 523379.72, 209099.85, -759153.26, …
$ PROX_PRIMARY_SCH.1      <dbl> 242668.03, 1106830.66, 571462.33, 3127477.21, …
$ PROX_SHOPPING_MALL.1    <dbl> 300881.390, -87693.378, -126732.712, -29593.34…
$ PROX_BUS_STOP.1         <dbl> 1210615.44, 1843587.22, 1411924.90, 7225577.51…
$ NO_Of_UNITS.1           <dbl> 104.8290640, -288.3441183, -9.5532945, -161.35…
$ FAMILY_FRIENDLY.1       <dbl> -9075.370, 310074.664, 5949.746, 1556178.531, …
$ FREEHOLD.1              <dbl> 303955.61, 396221.27, 168821.75, 1212515.58, 3…
$ y                       <dbl> 3000000, 3880000, 3325000, 4250000, 1400000, 1…
$ yhat                    <dbl> 2886531.8, 3466801.5, 3616527.2, 5435481.6, 13…
$ residual                <dbl> 113468.16, 413198.52, -291527.20, -1185481.63,…
$ CV_Score                <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ Stud_residual           <dbl> 0.38207013, 1.01433140, -0.83780678, -2.846146…
$ Intercept_SE            <dbl> 516105.5, 488083.5, 963711.4, 444185.5, 211962…
$ AREA_SQM_SE             <dbl> 823.2860, 825.2380, 988.2240, 617.4007, 1376.2…
$ AGE_SE                  <dbl> 5889.782, 6226.916, 6510.236, 6010.511, 8180.3…
$ PROX_CBD_SE             <dbl> 37411.22, 23615.06, 56103.77, 469337.41, 41064…
$ PROX_CHILDCARE_SE       <dbl> 319111.1, 299705.3, 349128.5, 304965.2, 698720…
$ PROX_ELDERLYCARE_SE     <dbl> 120633.34, 84546.69, 129687.07, 127150.69, 327…
$ PROX_URA_GROWTH_AREA_SE <dbl> 56207.39, 76956.50, 95774.60, 470762.12, 47433…
$ PROX_MRT_SE             <dbl> 185181.3, 281133.9, 275483.7, 279877.1, 363830…
$ PROX_PARK_SE            <dbl> 205499.6, 229358.7, 314124.3, 227249.4, 364580…
$ PROX_PRIMARY_SCH_SE     <dbl> 152400.7, 165150.7, 196662.6, 240878.9, 249087…
$ PROX_SHOPPING_MALL_SE   <dbl> 109268.8, 98906.8, 119913.3, 177104.1, 301032.…
$ PROX_BUS_STOP_SE        <dbl> 600668.6, 410222.1, 464156.7, 562810.8, 740922…
$ NO_Of_UNITS_SE          <dbl> 218.1258, 208.9410, 210.9828, 361.7767, 299.50…
$ FAMILY_FRIENDLY_SE      <dbl> 131474.73, 114989.07, 146607.22, 108726.62, 16…
$ FREEHOLD_SE             <dbl> 115954.0, 130110.0, 141031.5, 138239.1, 210641…
$ Intercept_TV            <dbl> 3.9720784, 3.3460017, 3.5629010, 0.5276150, 1.…
$ AREA_SQM_TV             <dbl> 11.614302, 20.087361, 13.247868, 33.577223, 4.…
$ AGE_TV                  <dbl> -1.6154474, -9.3441881, -4.1023685, -15.524301…
$ PROX_CBD_TV             <dbl> -3.22582173, -6.32792021, -4.62353528, 5.17080…
$ PROX_CHILDCARE_TV       <dbl> 1.000488185, 1.471786337, -0.344047555, 1.5766…
$ PROX_ELDERLYCARE_TV     <dbl> -3.26126929, 3.84626245, 4.13191383, 2.4756745…
$ PROX_URA_GROWTH_AREA_TV <dbl> -2.846248368, -1.848971738, -2.648105057, -5.6…
$ PROX_MRT_TV             <dbl> -1.61864578, -8.92998600, -3.40075727, -7.2870…
$ PROX_PARK_TV            <dbl> -0.83749312, 2.28192684, 0.66565951, -3.340617…
$ PROX_PRIMARY_SCH_TV     <dbl> 1.59230221, 6.70194543, 2.90580089, 12.9836104…
$ PROX_SHOPPING_MALL_TV   <dbl> 2.753588422, -0.886626400, -1.056869486, -0.16…
$ PROX_BUS_STOP_TV        <dbl> 2.0154464, 4.4941192, 3.0419145, 12.8383775, 0…
$ NO_Of_UNITS_TV          <dbl> 0.480589953, -1.380026395, -0.045279967, -0.44…
$ FAMILY_FRIENDLY_TV      <dbl> -0.06902748, 2.69655779, 0.04058290, 14.312764…
$ FREEHOLD_TV             <dbl> 2.6213469, 3.0452799, 1.1970499, 8.7711485, 1.…
$ Local_R2                <dbl> 0.8846744, 0.8899773, 0.8947007, 0.9073605, 0.…
$ coords.x1               <dbl> 22085.12, 25656.84, 23963.99, 27044.28, 41042.…
$ coords.x2               <dbl> 29951.54, 34546.20, 32890.80, 32319.77, 33743.…
$ geometry                <POINT [m]> POINT (22085.12 29951.54), POINT (25656.…
Show the code
summary(gwr.adaptive$SDF$yhat)
    Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
  171347  1102001  1385528  1751842  1982307 13887901 

Visualising local R2

Local R2 range between 0.0 and 1.0 and indicate how well the local regression model fits observed y values. Very low values indicate the local model is performing poorly. Mapping the Local R2 values to see where GWR predicts well and where it predicts poorly may provide clues about important variables that may be missing from the regression model.

Show the code
tmap_mode("view")
tmap mode set to interactive viewing
Show the code
tmap_options(check.and.fix = TRUE)
tm_shape(mpsz)+
  tm_polygons(alpha = 0.1) +
tm_shape(condo_resale.sf.adaptive) +  
  tm_dots(col = "Local_R2",
          border.col = "gray60",
          border.lwd = 1) +
  tm_view(set.zoom.limits = c(11,14))
Warning: The shape mpsz is invalid (after reprojection). See sf::st_is_valid
Show the code
tmap_mode("plot")
tmap mode set to plotting

By coefficient estimates

Show the code
tmap_mode("view")
tmap mode set to interactive viewing
Show the code
tmap_options(check.and.fix = TRUE)
AREA_SQM_SE <- tm_shape(mpsz)+
  tm_polygons(alpha = 0.1) +
tm_shape(condo_resale.sf.adaptive) +  
  tm_dots(col = "AREA_SQM_SE",
          border.col = "gray60",
          border.lwd = 1) +
  tm_view(set.zoom.limits = c(11,14))

AREA_SQM_TV <- tm_shape(mpsz)+
  tm_polygons(alpha = 0.1) +
tm_shape(condo_resale.sf.adaptive) +  
  tm_dots(col = "AREA_SQM_TV",
          border.col = "gray60",
          border.lwd = 1) +
  tm_view(set.zoom.limits = c(11,14))

tmap_arrange(AREA_SQM_SE, AREA_SQM_TV, 
             asp=1, ncol=2,
             sync = TRUE)
Warning: The shape mpsz is invalid (after reprojection). See sf::st_is_valid

Warning: The shape mpsz is invalid (after reprojection). See sf::st_is_valid
Show the code
tmap_mode("plot")
tmap mode set to plotting

By planning region

Show the code
tm_shape(mpsz[mpsz$REGION_N=="CENTRAL REGION", ])+
  tm_polygons()+
tm_shape(condo_resale.sf.adaptive) + 
  tm_bubbles(col = "Local_R2",
           size = 0.15,
           border.col = "gray60",
           border.lwd = 1)
Warning: The shape mpsz[mpsz$REGION_N == "CENTRAL REGION", ] is invalid. See
sf::st_is_valid